So I'm trying to solve the following exercise:
If $X$ is a normed space and $Y$ a finite dimensional subspace. Show that $Y$ is complemented in $X.$
I know there's a proof of it using the Hahn-Banach Theorem, however I'm trying to prove it without that. My approach is to use the fact that there exists $Z$ such that $Y\bigoplus Z = X$, thus use the projection $P$ onto $Y$, there is a theorem that says that $ Y$ is topologically complemented if and only if $P$ is bounded. I thought it would be easy to prove that $P$ is bounded but I can't actually prove it.